Mastering the technique of partial fractions is a cornerstone when integrating complex rational functions. This method simplifies these functions into a sum of simpler fractions that are more straightforward to integrate.

Decomposition into Partial Fractions

To integrate a complex rational function:

1. Break it down into simpler fractions with linear or quadratic denominators using partial fractions.

2. Integrate each simpler fraction separately.

Example:

For $\int_{1}^{2} \frac{2x^3 - 1}{x^2(2x - 1)} dx$:

Express as partial fractions:

Integration After Decomposition

Integrate a function by partial fractions:

1. Decompose into simpler fractions.

2. Integrate each fraction.

Example:

For $\int_{1}^{2} \frac{2x^3 - 1}{x^2(2x - 1)} dx$:

Decomposed form:

Integrated result:

With limits applied: